COMPUTATIONAL METHODS for ELECTRIC POWER SYSTEMS SECOND EDITION The ELECTRIC POWER ENGINEERING Series Series Editor Leo L Grigsby Published Titles Computational Methods for Electric Power Systems, Second Edition Mariesa L Crow Electric Energy Systems: Analysis and Operation Antonio Gómez-Expósito, Antonio J Conejo, and Claudio Cañizares Distribution System Modeling and Analysis, Second Edition William H Kersting Electric Machines Charles A Gross Harmonics and Power Systems Francisco C De La Rosa Electric Drives, Second Edition Ion Boldea and Syed Nasar Power System Operations and Electricity Markets Fred I Denny and David E Dismukes Power Quality C Sankaran Electromechanical Systems, Electric Machines,and Applied Mechatronics Sergey E Lyshevski Linear Synchronous Motors: Transportation and Automation Systems Jacek Gieras and Jerry Piech Electrical Energy Systems, Second Edition Mohamed E El-Hawary The Induction Machine Handbook Ion Boldea and Syed Nasar Electric Power Substations Engineering John D McDonald Electric Power Transformer Engineering James H Harlow Electric Power Distribution Handbook Tom Short SECOND EDITION COMPUTATIONAL METHODS for ELECTRIC POWER SYSTEMS MARIESA L CROW Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number-13: 978-1-4200-8661-4 (Ebook-PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and 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registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To Jim, David, and Jacob Preface to the Second Edition This new edition has been updated to include new material Specifically, this new edition has added sections on the following material: • Generalized Minimal Residual (GMRES) methods • Numerical differentiation • Secant method • Homotopy and continuation methods • Power method for computing dominant eigenvalues • Singular-value decomposition and pseudoinverses • Matrix pencil method and a significant revision of the Optimization chapter (Chapter 6) to include linear and quadratic programming methods A course structure would typically include the following chapters in sequence: Chapter 1, 2, and From this point, any of the chapters can follow without loss of consistency I have tried to structure each chapter to give the reader an overview of the methods with salient examples In many cases however, it is not possible to give an exhaustive coverage of the material; many topics have decades of work devoted to their development Many of the methods presented in this book have commercial software packages that will accomplish their solution far more rigorously with many failsafe attributes included (such as accounting for ill-conditioning, etc.) It is not my intent to make students experts in each topic, but rather to develop an appreciation for the methods behind the packages Many commercial packages provide default settings or choices of parameters for the user; through better understanding of the methods driving the solution, informed users can make better choices and have a better understanding of the situations in which the methods may fail If this book provides any reader with more confidence in using commercial packages, I have succeeded in my intent As before, I am indebted to many people: my husband Jim and my children David and Jacob for making every day a joy, my parents Lowell and Sondra for their continuing support, and Frieda Adams for all she does to help me succeed Mariesa L Crow Rolla, Missouri 2009 Preface to the First Edition This book is the outgrowth of a graduate course that I’ve taught at the University of Missouri-Rolla for the past decade or so Over the years, I’ve used a number of excellent textbooks for this course, but each textbook was always missing some of the topics that I wanted to cover in the class After relying on handouts for many years, my good friend Leo Grigsby encouraged me to put them down in the form of a book (if arm-twisting can be called encouragement ) With the support of my graduate students, who I used as testbeds for each chapter, this book gradually came into existence I hope that those who read this book will find this field as stimulating as I have found it In addition to Leo and the fine people at CRC Press, I’m grateful to the University of Missouri-Rolla administration and the Department of Electrical and Computer Engineering for providing the environment to nurture my teaching and research and giving me the latitude to pursue my personal interests in this field Lastly, I don’t often get the opportunity to publicly acknowledge the people who’ve been instrumental in my professional development I’d like to thank: Marija Ilic, who initially put me on the path; Peter Sauer, who encouraged me along the way; Jerry Heydt, for providing inspiration; Frieda Adams, for all she does to make my life easier; Steve Pekarek, for putting up with my grumbling and complaining; and Lowell and Sondra Crow for making it all possible Mariesa L Crow Rolla, Missouri 2003 Eigenvalue Problems 275 0.9 0.8 0.7 |Y(f)| 0.6 0.5 0.4 0.3 0.2 0.1 0 10 15 20 25 30 Frequency (rad) 35 40 45 50 FIGURE 7.4 Frequency response of Figure 7.3 Levenberg-Marquardt mode bi ωi θi 1.0028 -0.0110 7.9998 0.0014 0.6010 -0.0305 16.9994 3.1426 0.5051 0.0378 4.6967 0.7989 The reconstruction error in each waveform is measured N m ak e(bk ti ) cos (ωk ti + θk ) − yi error = i=1 (7.65) k=1 and the errors for each method are: Method error Matrix Pencil 0.1411 Levenberg-Marquardt 0.0373 Prony 3.9749 Not surprisingly, the Levenberg-Marquardt yielded the best results since it is an iterative method, whereas the other estimation methods are linear non-iterative methods Power System Example In this example, the accuracy of the methods will be compared using the dynamic response of PSS/E simulation of a large Midwestern utility system shown in Figure 7.5 This simulation contains several hundred states comprising a wide range of responses The number of dominant modes is not known 276 Computational Methods for Electric Power Systems 80 79 78 signal 77 76 75 74 73 10 time (seconds) FIGURE 7.5 PSS/E waveform 0.9 0.8 0.7 |Y(f)| 0.6 0.5 0.4 0.3 0.2 0.1 0 10 15 Frequency (rad) FIGURE 7.6 FFT of PSS/E waveform 20 25 Eigenvalue Problems 277 80 Original Levenberg−Marquardt Matrix Pencil Prony 79 78 x(t) 77 76 75 74 73 time (seconds) 10 FIGURE 7.7 Reconstruction of the PSS/E waveform using various methods Using the Hilbert method first to determine the number of probable modes yields the FFT results shown in Figure 7.6 From this figure, it appears as if there are five dominant modes that contribute significantly to the original waveform, with several of the modes concentrated at low frequencies Therefore, the estimation methods introduced earlier will be applied to extract five modes Extracting five modes, the results are shown in Figure 7.7 and summarized: Prony: mode 1.7406 1.5723 1.0504 2.1710 0.9488 bi -0.5020 -0.1143 -0.0156 -0.2455 -0.3515 ωi 3.7835 4.8723 6.2899 7.7078 8.3854 θi -1.4870 -1.1219 -0.0331 2.2011 -1.6184 mode 2.0317 1.3204 0.7035 1.2935 0.6718 bi -0.5610 -0.0774 0.0527 -0.2400 -0.0826 ωi θi 3.7357 -1.5158 4.8860 -1.3607 6.3030 -0.3093 7.4175 2.9957 8.0117 0.3790 Matrix Pencil 278 Computational Methods for Electric Power Systems Levenberg-Marquardt mode 1.8604 1.1953 0.8164 1.9255 0.6527 bi -0.5297 -0.0578 0.0242 -0.2285 -0.2114 ωi 3.6774 4.8771 6.2904 7.6294 8.3617 θi -1.3042 -1.3405 -0.2537 2.1993 -1.5187 The error in each method as determined by equation (7.65) and the relative computation times are: method CPU Prony 0.068 Matrix Pencil 39.98 Levenberg-Marquardt 42.82 Error 228.16 74.81 59.74 Depends on initial condition Note that the Prony method is the most computationally efficient since it only requires two least squares solutions The Matrix Pencil method is more computationally expensive since it requires a singular value decomposition of a relatively large matrix It also requires an eigensolution, but since the matrix itself is relatively small (the size of the number of required modes), it is not overly burdensome Not surprisingly, the Levenberg-Marquardt method is the most computationally expensive since it is an iterative method Its computational burden is directly related to the initial guess: the better the initial guess, the faster the method converges It is wise to choose the initial guess as the parameters obtained from either the Prony or the Matrix Pencil methods Similarly, the level of error in each method varies with the complexity of the method The Levenberg-Marquardt method yields the best results, but with the greatest computational effort The Prony has the largest error, but this is offset by the relative speed of computation 7.6 7.6.1 Power System Applications Participation Factors In the analysis of large scale power systems, it is sometimes desirable to have a measure of the impact that a particular state has on a selected system mode (or eigenvalue) In some cases, it is desirable to know whether a set of physical states has influence over an oscillatory mode such that control of Eigenvalue Problems 279 that component may mitigate the oscillations Another use is to identify which system components contribute to an unstable mode One tool for identifying which states significantly participate in a selected mode is the method of participation factors [57] In large scale power systems, participation factors can also be used to identify inter-area oscillations versus those that persist only within localized regions (intra-area oscillations) Participation factors provide a measure of the influence each dynamic state has on a given mode or eigenvalue Consider a linear system x˙ = Ax (7.66) The participation factor pki is a sensitivity measure of the ith eigenvalue to the (k, k) diagonal entry of the system A matrix This is defined as pki ∂λi ∂akk (7.67) where λi is the ith eigenvalue and akk is the k th diagonal entry of A The participation factor pki relates the k th state variable to the ith eigenvalue An equivalent, but more common expression for the participation factor is also defined as wki vik pki = T (7.68) wi vi where wki and vki are the k th entries of the left and right eigenvectors associated with λi As with eigenvectors, participation factors are frequently normalized to unity, such that n pki = (7.69) k=1 When the participation factors are normalized, they provide a straightforward measure of the percent of impact each state has on a particular mode Participation factors for complex eigenvalues (and eigenvectors) are defined in terms of magnitudes, rather than complex quantities In the case of complex eigenvalues, the participation factors are defined pki = |vik ||wki | n i=1 |vik ||wki | (7.70) In some applications, it may be preferred to retain the complex nature of the participation factors to yield both phase and magnitude information [29] 280 Computational Methods for Electric Power Systems 2.5 1.5 x(t) 0.5 −0.5 −1 −1.5 −2 −2.5 time (seconds) 10 FIGURE 7.8 Waveform for Problem 7.7 Problems Find the eigenvalues and eigenvectors of the following matrices ⎤ ⎡ 5411 ⎢4 1⎥ ⎥ A1 = ⎢ ⎣1 2⎦ 1124 ⎤ ⎡ 34 A2 = ⎣ −1 ⎦ −1 Find the eigenvalues of the follow matrix using the shifted-QR method ⎤ ⎡ 001 A = ⎣1 0⎦ 010 Generate the waveform shown in Figure 7.8 on the interval t ∈ [0, 10] with a time step of 0.01 seconds ebi t (cos ci t + di ) x(t) = i=1 Eigenvalue Problems 281 where mode 1.0 0.6 0.5 bi ci di -0.01 8.0 0.0 -0.03 17.0 π 0.04 4.7 π/4 (a) Using 100 equidistant points on the interval [0, 10], estimate the six system eigenvalues using Prony analysis How these compare with the actual eigenvalues? (b) Using 100 equidistant points on the interval [0, 10], estimate the six system eigenvalues using Levenberg-Marquardt How these eigenvalues compare with the actual eigenvalues? with those obtained from the Prony analysis? (c) Using 100 equidistant points on the interval [0, 10], estimate the six system eigenvalues using the matrix pencil method How these eigenvalues compare with the actual eigenvalues? with those obtained from the Prony analysis? (d) Using all of the points, estimate the six system eigenvalues using Prony analysis How these compare with the actual eigenvalues? (e) Using all of the points, estimate the six system eigenvalues using Levenberg-Marquardt How these compare with the actual eigenvalues? (f) Using all of the points, estimate the six system eigenvalues using the matrix pencil method How these compare with the actual eigenvalues? 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POWER SYSTEMS SECOND EDITION The ELECTRIC POWER ENGINEERING Series Series Editor Leo L Grigsby Published Titles Computational Methods for Electric Power Systems, ... L11 1 (b2 − L21 y1 ) (1 − (2 )(1 )) = = L22 −5 (b3 − (L31 y1 + L32 y2 )) = ( 5) − (4 )(1 ) + ( 9) L33 (b4 − (L41 y1 + L42 y2 + L43 y3 )) L44 − (9 )(1 ) + ( 25) 15 + (1 )(6 ) = −6 Thus =6 ⎡ ⎤ ⎢1⎥ ⎢ y=⎣5⎥... true for both column and 14 Computational Methods for Electric Power Systems row calculations The second row of Q is computed: (a23 − q21 q13 ) = q22 = (a24 − q21 q14 ) = q22 q23 = q24 (2 − (2 )(4 ))